Abstract
In situations with incomplete information, we may have only partial knowledge about a concept. This motivates the introduction of the notion of a partially-known concept represented by a set of known instances, a set of known non-instances, and a set of objects with unknown states. We present a common conceptual framework of the notions of interval sets and incomplete formal contexts for representing partially-known concepts. An interval set is interpreted as a family of sets bounded by a pair of sets, and any one in the family may possibly be the actual set of instances of the concept when the information or knowledge becomes complete. An incomplete formal context is interpreted as a family of complete formal contexts, and any one in the family may possibly be the actual formal context when the information or knowledge becomes complete. While a complete formal context is induced by a binary relation, an incomplete formal context is induced by an interval binary relation that is interpreted as a family of binary relations. Within the proposed framework, we identify four possible forms for representing partially-known concepts. We examine, interpret, and extend existing studies on concept analysis in complete formal contexts.
Notes
The notions of the least and the greatest completions used here and ill-known formal concepts used later are taken from two English presentations: http://www.loria.fr/~kaytouem/treillis-clermontois/treillis09_Assaghir.pdf (Accessed June 15, 2016), https://www.polytech.univ-savoie.fr/fileadmin/polytech_autres_sites/sites/lfa2009/templates/presentation/representation_connaissance/Djouadi.pdf (Accessed June 8, 2016).
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Acknowledgments
This work was supported in part by a Discovery Grant from NSERC, Canada. The author thanks Dr. Weihua Xu, Dr. Jinhai Li and Dr. Mingwen Shao for their encouragement during the preparation of the paper. The author is grateful to Dr. Jianjun Qi, Dr. Ling Wei, Ruisi Ren, and Mengjun Hu, and anonymous reviewers for their constructive comments.
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Yao, Y. Interval sets and three-way concept analysis in incomplete contexts. Int. J. Mach. Learn. & Cyber. 8, 3–20 (2017). https://doi.org/10.1007/s13042-016-0568-1
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DOI: https://doi.org/10.1007/s13042-016-0568-1